Found problems: 1
2024 Brazil Cono Sur TST, 2
Inside an angle $\angle BOC$ there are three disjoint circles: $k_1,k_2$ and $k_3$, which are, each one, tangent to its sides $BO$ and $OC$. Let $r_1, r_2$ and $r_3$, respectively, be the radii of these circles, with $r_1<r_2<r_3$. The circles $k_1$ and $k_3$ are tangent to the side $OB$ at $A$ and $B$, respectively, and $k_2$ is tangent to the side $OC$ at $C$. Let $K=AC\cap k_1,L=AC\cap k_2,M=BC\cap k_2$ and $N=BC\cap k_3$. Besides that, let $P=AM\cap BK,Q=AM\cap BL,R=AN\cap BK$ and $S=AN\cap BL$. If the intersections of $CP,CQ,CR$ and $CS$ with $AB$ are $X,Y,Z$ and $T$, respectively, prove that $XZ = YT$.